Hi, recently I've been researching path synthesis in aircraft navigation and stumbled across a few images of constructions used to join different navigation paths together. In the image linked above, I will know the values of c,g,p1,p2&p3 and will need to find points c1,i1 and i2. I first tried to solve this by going over a few circle problems from my school days but always ended up at dead ends, for example, I tried computing the length between the center of the two circles and making the resulting equation equal to the sum of the two radii(touching circles). This left me with three unknowns obviously and I wasn't sure how to proceed, I also wondered if the tangent |P1P2| would be of any help with this problem. This is as far as I've gotten unfortunately and any help to point me in the right direction would be greatly appreciated.
Knowing $r$, draw the two dashed lines (circle centered at $c$, radius $g+r$ and parallel to $p_1p_2$ at distance $r$). This gives you $c_1$.
If you want an analytical solution, solve
$$\begin{cases}(x-x_c)^2+(y-y_c)^2&=(g+r)^2,\\y&=y_1+r\end{cases}$$
or
$$x=x_c+\sqrt{(g+r)^2-(y_1+r-y_c)^2}.$$